# Why do the gross return of wealth and wealth move in opposite direction?

I have a short question:

Why does the gross return of wealth decrease as wealth increases?

Why do the gross return of wealth and wealth move in opposite direction?

$$R=1+(a/w^2)$$ where a is non-negative parameter.

• Are you sure your expression is correct? I believe you should have a minus sign in there, not a plus. – heh Nov 20 at 18:41
• In the question, this equation is given like that. So I don't know. If what you said is true, please can you refer it by a book? @heh – user315 Nov 20 at 18:56
• Do you know calculus? – heh Nov 20 at 19:19
• @heh yes I know of course – user315 Nov 20 at 20:52
• OK: I'll write out an answer below that should help. – heh Nov 20 at 20:55

Gross return can be written as $$R = 1 + \frac{a}{w}$$ where $$w$$ is initial wealth (or more accurately, the initial investment) and $$a$$ is new cash flow generated by the investment. So for example, with $$w = 100$$ and $$a = 5$$, the gross return is $$R = 1 + \frac{5}{100}$$ = 1.05, representing a 5% return on \$100.
What you call the "gross return of wealth" is, I think, just the derivative of $$R$$ with respect to $$w$$: $$\frac{dR}{dw} = -\frac{a}{w^2}$$. The question is, why is this negative? Because what this quantity tells you is how your gross return changes when your initial investment grows, holding the new cash flow constant.
It really just follows from the arithmetic of proportions. Return is a proportion while wealth is a number. As wealth $$w$$ increases, the return from a new cash flow $$a$$ is reduced, because it is a smaller proportion of a larger whole. So in the example above, if we double our investment to $$w = 200$$ but we keep $$a = 5$$, then the return is $$R = 1 + \frac{5}{200} = 1.025$$. Because we had to invest more to get the same amount of new dollars, our return is smaller.